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Find the roots of the quadratic equations by applying the quadratic formula.
$4x^2 - 4\sqrt3x + 3 = 0$
Given:
Given quadratic equation is $4x^2 - 4\sqrt3x + 3 = 0$
To do:
We have to find the roots of the given quadratic equation.
Solution:
$4x^2 - 4\sqrt3x + 3 = 0$
The above equation is of the form $ax^2 + bx + c = 0$, where $a = 4, b = -4\sqrt3$ and $c = 3$
Discriminant $\mathrm{D} =b^{2}-4 a c$
$=(4 \sqrt{3})^{2}-4 \times 4 \times 3$
$=48-48$
$=0$
$\mathrm{D}=0$
Let the roots of the equation are $\alpha$ and $\beta$
$\alpha=\frac{-b+\sqrt{\mathrm{D}}}{2 a}$
$=\frac{-4 \sqrt{3}+0}{8}$
$=\frac{-4 \sqrt{3}}{8}$
$=\frac{-\sqrt{3}}{2}$
$\beta=\frac{-b-\sqrt{\mathrm{D}}}{2 a}$
$=\frac{-4 \sqrt{3}-0}{8}$
$=\frac{-4 \sqrt{3}}{8}$
$=\frac{-\sqrt{3}}{2}$
Hence, the roots of the given quadratic equation are $\frac{-\sqrt{3}}{2}, \frac{-\sqrt{3}}{2}$.
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